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Chapter 12

Q. 12.5

A plane layer of gray medium with thickness D and constant properties is initially at uniform temperature T_0. It has attenuation coefficient β, density ρ, and specific heat c_υ. At time t = 0 the layer is subjected to a very cold environment. Consider only radiation transfer and obtain the transient temperature of the layer by using the emission approximation.

Step-by-Step

Verified Solution

From the results of Example 12.4, the instantaneous transient heat flux emerging from both boundaries of the layer is q(t=)4\beta \sigma T^4(t)D.The energy equation for the layer then becomes ρc_υdT /dt=-4\beta \sigma T^4(t),or, in dimensionless form,d\vartheta /\overline{dt} =-4\vartheta ^4(\overline{t} ) where ϑ=T/T_0 and \overline{t}=(\beta \sigma T_0^3/\rho c_v)t. Integrating with the condition that ϑ = 1 at \overline{t}=0 gives the transient uniform temperature throughout the layer for the emission approximation as

\vartheta (\overline{t} )=\frac{1}{(1+2\overline{t})^{1/3}}